To solve the problem step by step, we will calculate the increase in mass of the copper when it is heated from 0 to 1000 °C.
### Step 1: Understand the relationship between heat, mass, specific heat, and temperature change.
The heat \( Q \) absorbed or released by a substance can be calculated using the formula:
\[
Q = mc\Delta T
\]
where:
- \( m \) = mass of the substance (in kg)
- \( c \) = specific heat capacity (in J/kg·K)
- \( \Delta T \) = change in temperature (in K or °C)
### Step 2: Identify the given values.
From the problem, we have:
- Mass of copper, \( m = 2 \, \text{kg} \)
- Specific heat of copper, \( c = 378 \, \text{J/kg·K} \)
- Change in temperature, \( \Delta T = 1000 \, \text{°C} \)
### Step 3: Calculate the heat absorbed by the copper.
Using the formula for heat:
\[
Q = mc\Delta T
\]
Substituting the values:
\[
Q = 2 \, \text{kg} \times 378 \, \text{J/kg·K} \times 1000 \, \text{K}
\]
\[
Q = 2 \times 378 \times 1000 = 756000 \, \text{J}
\]
### Step 4: Use Einstein's equation to find the increase in mass.
According to Einstein's mass-energy equivalence principle:
\[
E = \Delta m c^2
\]
Rearranging to find the change in mass \( \Delta m \):
\[
\Delta m = \frac{E}{c^2}
\]
where \( E \) is the energy (heat) calculated in the previous step, and \( c \) is the speed of light (\( c \approx 3 \times 10^8 \, \text{m/s} \)).
### Step 5: Substitute the values into the equation.
First, calculate \( c^2 \):
\[
c^2 = (3 \times 10^8 \, \text{m/s})^2 = 9 \times 10^{16} \, \text{m}^2/\text{s}^2
\]
Now substitute \( E \) and \( c^2 \) into the equation for \( \Delta m \):
\[
\Delta m = \frac{756000 \, \text{J}}{9 \times 10^{16} \, \text{m}^2/\text{s}^2}
\]
Calculating \( \Delta m \):
\[
\Delta m = \frac{756000}{9 \times 10^{16}} \approx 8.4 \times 10^{-12} \, \text{kg}
\]
### Final Answer:
The increase in mass of the copper is approximately \( 8.4 \times 10^{-12} \, \text{kg} \).
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